Based on the van der Waals (vdW) interaction potential and the bending rigidity of graphene, the self-scrolling mechanical behaviors of monolayer graphene flake induced by carbon nanotube (CNT)/black phosphorus nanotube (BPNT) are studied, and the analytical expressions of critical radius of CNT/BPNT are derived when the graphene flakes spontaneously scroll round them. The factors affecting the critical radius of nanotubes are systematically analyzed, which include the axial length ratio between nanotube and graphene, wrapping angle of graphene, temperature, type and layer number of nanotube. It is found that the axial length ratio between nanotube and graphene has a significant effect on the critical radius. The critical radius of nanotubes increases with the decrease of the axial length ratio, and the critical radius of double-walled carbon nanotube (DWCNT) increases faster than that of single-walled carbon nanotube (SWCNT) and BPNT. When the SWCNT and graphene have the same axial length, the critical radius of SWCNT induced completely wrapping of monolayer graphene is about 0.4757 nm, which is in good agreement with the simulation results in the literature. The wrapping angle of graphene can also affect the critical radius of nanotubes, which increases with the increase of wrapping angle. Furthermore, the influence of environmental temperature, the type and layer number of nanotubes on the critical radius of nanotubes is discussed, respectively.

中文翻译：

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纳米管诱导石墨烯薄片自卷曲临界半径的理论研究

基于范德华（vdW）相互作用势和石墨烯的弯曲刚度，研究了碳纳米管（CNT）/黑磷纳米管（BPNT）诱导的单层石墨烯薄片自卷曲力学行为，并得到了解析表达式CNT/BPNT 的临界半径是在石墨烯薄片自发地围绕它们滚动时得出的。系统分析了影响纳米管临界半径的因素，包括纳米管与石墨烯的轴长比、石墨烯的包角、温度、纳米管的类型和层数。发现纳米管和石墨烯之间的轴长比对临界半径有显着影响。纳米管的临界半径随着轴长比的减小而增大，双壁碳纳米管（DWCNT）的临界半径比单壁碳纳米管（SWCNT）和BPNT增加得更快。当单壁碳纳米管和石墨烯具有相同的轴向长度时，单壁碳纳米管诱导单层石墨烯完全包裹的临界半径约为0.4757 nm，这与文献中的模拟结果非常吻合。石墨烯的包角也会影响纳米管的临界半径，随着包角的增加而增加。此外，分别讨论了环境温度、纳米管类型和层数对纳米管临界半径的影响。SWCNT诱导单层石墨烯完全包裹的临界半径约为0.4757 nm，与文献中的模拟结果非常吻合。石墨烯的包角也会影响纳米管的临界半径，随着包角的增加而增加。此外，分别讨论了环境温度、纳米管类型和层数对纳米管临界半径的影响。SWCNT诱导单层石墨烯完全包裹的临界半径约为0.4757 nm，与文献中的模拟结果非常吻合。石墨烯的包角也会影响纳米管的临界半径，随着包角的增加而增加。此外，分别讨论了环境温度、纳米管类型和层数对纳米管临界半径的影响。